Download Computing prime factors with a Josephson phase qubit quantum

Computing prime factors with a Josephson phase qubit quantum processor
Erik Lucero,1 R. Barends,1 Y. Chen,1 J. Kelly,1 M. Mariantoni,1 A. Megrant,1 P. O’Malley,1 D.
Sank,1 A. Vainsencher,1 J. Wenner,1 T. White,1 Y. Yin,1 A. N. Cleland,1 and John M. Martinis1
1
Department of Physics, University of California, Santa Barbara, CA 93106, USA
A quantum processor (QuP) can be used to exploit quantum mechanics to find the prime factors
of composite numbers[1]. Compiled versions of Shor’s algorithm have been demonstrated on ensemble quantum systems[2] and photonic systems[3–5], however this has yet to be shown using solid
state quantum bits (qubits). Two advantages of superconducting qubit architectures are the use of
conventional microfabrication techniques, which allow straightforward scaling to large numbers of
qubits, and a toolkit of circuit elements that can be used to engineer a variety of qubit types and
interactions[6, 7]. Using a number of recent qubit control and hardware advances [7–13], here we
demonstrate a nine-quantum-element solid-state QuP and show three experiments to highlight its
capabilities. We begin by characterizing the device with spectroscopy. Next, we produces coherent
interactions between five qubits and verify bi- and tripartite entanglement via quantum state tomography (QST) [8, 12, 14, 15]. In the final experiment, we run a three-qubit compiled version of
Shor’s algorithm to factor the number 15, and successfully find the prime factors 48 % of the time.
Improvements in the superconducting qubit coherence times and more complex circuits should provide the resources necessary to factor larger composite numbers and run more intricate quantum
algorithms.
In this experiment, we scaled-up from an architecture initially implemented with two qubits and three resonators [7] to a nine-element quantum processor (QuP)
capable of realizing rapid entanglement and a compiled
version of Shor’s algorithm. The device is composed
of four phase qubits and five superconducting coplanar
waveguide (CPW) resonators, where the resonators are
used as qubits by accessing only the two lowest levels.
Four of the five CPWs can be used as quantum memory elements as in Ref. [7] and the fifth can be used to
mediate entangling operations.
The QuP can create entanglement and execute quantum circuits[16, 17] with high-fidelity single-qubit gates
(X, Y , Z, and H), [18, 19]combined with swaps and
controlled-phase (Cφ ) gates[7, 13, 20], where one qubit interacts with a resonator at a time. The QuP can also utilize “fast-entangling logic” by bringing all participating
qubits on resonance with the resonator at the same time
to generate simultaneous entanglement[21]. At present,
this combination of entangling capabilities has not been
demonstrated on a single device. Previous examples have
shown: spectroscopic evidence of the increased coupling
for up to three qubits coupled to a resonator[14], as well
as coherent interactions between two and three qubits
with a resonator[12], although these lacked tomographic
evidence of entanglement.
Here we show coherent interactions for up to four
qubits with a resonator and verify genuine bi- and tripartite entanglement including Bell [9] and |Wi-states [10]
with quantum state tomography (QST). This QuP has
the further advantage of creating entanglement at a rate
more than twice that of previous demonstrations[10, 12].
In addition to these characterizations, we chose to implement a compiled version of Shor’s algorithm[22, 23],
in part for its historical relevance[16] and in part because
this algorithm involves the challenge of combining both
single- and coupled-qubit gates in a meaningful sequence.
We constructed the full factoring sequence by first performing automatic calibration of the individual gates and
then combined them, without additional tuning, so as to
factor the composite number N = 15 with co-prime a = 4,
(where 1 < a < N and the greatest common divisor between a and N is 1). We also checked for entanglement
at three points in the algorithm using QST.
Figure 1a shows a micrograph of the QuP, made on
a sapphire substrate using Al/AlOx /Al Josephson junctions. Figure 1b shows a complete schematic of the device. Each qubit Qi is individually controlled using a bias
coil that carries dc, rf- and GHz-pulses to adjust the qubit
frequency and to pulse microwaves for manipulating and
measuring the qubit state. Each qubit’s frequency can be
adjusted over an operating range of ∼ 2 GHz, allowing us
to couple each qubit to the other quantum elements on
the chip. Each Qi is connected to a memory resonator
Mi , as well as the bus B, via interdigitated capacitors.
Although the coupling capacitors are fixed, Fig. 1c illustrates how the effective interaction can be controlled by
tuning the qubits into or near resonance with the coupling bus (coupling “on”) or detuning Qi to fB ±500 MHz
(coupling “off”)[24].
The QuP is mounted in a superconducting aluminum
sample holder and cooled in a dilution refrigerator to
∼ 25 mK. Qubit operation and calibration are similar
to previous works[7–10, 13],with the addition of an automated calibration process[25]. As shown in Fig.1d, we
used swap spectroscopy[7] to calibrate all nine of the engineered quantum elements on the QuP, the four phase
qubits (Q1 − Q4 ), the four quarter-wave CPW quantum
memory resonators (M1 − M4 ), and one half-wave CPW
bus resonator (B). The coupling strengths between Qi
2
a
M2
Q2
Co
l
ro
nt Q
D
UI
SQ
M4
1
B
M1
Q4
M3
b
Q2
1mm
Q3
M2
M4
Q4
M3
Q3
λ/2
dc, rf, GHz
Control
B
Q1
SQUID
M1
λ/4
c
M2
frequency
M1
rf
M3
Q1
Q2
Q4
B
d
M4
Q3
Q1
Q2
Q3
7.2
M1
Q4
M3
M2
M4
6.9
frequency [GHz]
6.8
B
6.1
0
200
Δτ (ns)
0
0.0
200
Δτ (ns)
0
Pe
200
Δτ (ns)
0
200
Δτ (ns)
1.0
FIG. 1: Architecture and operation of the quantum processor (QuP). a, Photomicrograph of the sample, fabricated
with aluminum (colored) on sapphire substrate (dark). b,
Schematic of the QuP. Each phase qubit Qi is capacitively
coupled to the central half-wavelength bus resonator B and a
quarter-wavelength memory resonator Mi . The control lines
carry GHz microwave pulses to produce single qubit operations. Each Qi is coupled to a superconducting quantum
interference device (SQUID) for single-shot readout. c, Illustration of QuP operation. By applying pulses on each control
line, each qubit frequency is tuned in and out of resonance
with B (M) to perform entangling (memory) operations. d,
Swap spectroscopy[7] for all four qubits: Qubit excited state
|ei probability Pe (color scale) versus frequency (vertical axis)
and interaction time ∆τ . The centers of the chevron patterns gives the frequencies of the resonators B, M1 − M4 ,
f = 6.1, 6.8, 7.2, 7.1, 6.9 GHz respectively. The oscillation periods give the coupling strengths between Qi and B (Mi ),
which are all ∼
= 55 MHz (∼
= 20 MHz).
and B (Mi ) were measured to be within 5 % (10 %) of
the design values.
The qubit-resonator interaction can be described by
the
model Hamiltonian[26] Hint =
P Jaynes-Cummings
+
† −
(h̄g
/2)(a
σ
+aσ
),
where gi is the coupling strength
i
i
i
i
between the bus resonator B and the qubit Qi , a† and
a are respectively the photon creation and annihilation operators for the resonator, σi+ and σi− are respectively the qubit Qi raising and lowering operators, and
h̄ = h/2π. The dynamics during the interaction between
the i = {1, 2, 3, 4} qubits and the bus resonator are shown
in Fig.1c, and Fig.2 a, b, c respectively.
For these interactions the qubits Q1 −Q4 are initialized
in the ground state |ggggi and tuned off-resonance from
B at an idle frequency f ∼ 6.6 GHz. Q1 is prepared in
the excited state |ei via a π-pulse. B is then pumped
into the n = 1 Fock state[8] by tuning Q1 on resonance
(f ∼ 6.1 GHz) for a duration 1/2g1 = τ ∼ 9 ns, long
enough to complete an iSWAP operation between Q1 and
B, |0i ⊗ |egggi → |1i ⊗ |ggggi.
The participating qubits are then tuned on resonance
(f ∼ 6.1 GHz) and left to interact with B for an interaction time ∆τ . Figures 2 a,b,c show the probability PQi of
measuring the participating qubits in the excited state,
and the probability PB of B being in the n = 1 Fock
state, versus ∆τ . At the beginning of the interaction the
excitation is initially concentrated in B (PB maximum)
then spreads evenly between the participating qubits (PB
minimum) and finally returns back to B, continuing as a
coherent oscillation during this interaction time.
When the qubits are simultaneously tuned on resonance with B they interact with an effective coupling
√
strength ḡN that scales with number of qubits as N [14],
analogous to a single qubit coupled to a resonator
in a
√
N-photon Fock
state[8].
For
N
qubits,
ḡ
=
N
ḡ,
where
N
P
ḡ = [1/N ( i=1,N gi2 )]1/2 . The oscillation frequency of
PB for each of the four cases i = {1, 2, 3, 4} is shown in
Fig.2 d. These results are similar to Ref.[14], but with
a larger number N of qubits
√ interacting with the resonator, we can confirm the N scaling of the coupling
strength with N. From these data we find a mean value
of ḡ = 56.5 ± 0.05 MHz.
By tuning the qubits on resonance for a specific interaction time τ , corresponding to the first minimum of PB
in Fig.√2a, b, we can generate Bell singlets |ψS i = (|gei
√−
|egi)/ 2 and |Wi states |Wi = (|ggei+|gegi+|eggi)/ 3.
Stopping the interaction at this time (τBell = 6.5 ns
and τW = 5.1 ns) leaves the single excitation evenly distributed among the participating qubits and places the
qubits in the desired equal superposition state similar to
the protocol in Ref.[12]. We are able to further analyze
these states using full QST. Figures 2e, f show the reconstructed density matrices from this analysis[15]. The Bell
singlet is formed with fidelity FBell = hψs | ρBell |ψs i =
0.89 ± 0.01 and entanglement of formation[27] EOF =
0.70. The three-qubit |Wi state is formed with fidelity
1.0 a
c
d
e
N=3
0.5
a
N=2
0
Q4
Q2
b
1/2
B
4
0
1.0
ψs
Q3
Q1
0.5
0
1.0
PQ1,PQ4,PB
N=4
Number of qubits (N)
PQ1,PQ2,PQ4,PB
PQ1,PQ2,PQ3,PQ4,PB
3
Q1
B
ee
3
Q2
-1/2
ee
gg
gg
Q4
f
W
Q1
1/3
B
2
Q4
Q1
0.5
B
1
0
eee
ggg
eee
ggg
0
0
20
40
60
80
Interaction time Δτ (ns)
100
50
70
90
110
130
Oscillation frequency (MHz)
FIG. 2: Rapid entanglement for two to four-qubits. Panels a,b,c show the measured state occupation probabilities PQ1−4 (color)
and PB (black) for increasing number of participating qubits N = {2, 3, 4} versus interaction time ∆τ . In all cases B is first
prepared in the n = 1 Fock state[8] and the participating qubits are then tuned on resonance with B for the interaction time ∆τ .
The single excitation begins in B, spreads to the participating qubits, and then returns to B. These coherent oscillations continue
for a time ∆τ and increase in frequency with each additional qubit. d, Oscillation frequency of PB for increasing numbers
of participating qubits. The error bars indicate the −3 dB point of the Fourier
√ transformed PB data. The inset schematics
illustrate which qubits participate. The coupling strength increases as ḡN = N ḡ, plotted as a black line fit to the data, with
√
ḡ = 56.5 ± 0.05 MHz. e,f The real part of the reconstructed density matrices from QST. (e), Bell singlet |ψs i = (|gei
√ − |egi)/ 2
with fidelity FBell = hψs | ρBell |ψs i = 0.89±0.01 and EOF = 0.70. (f), Three-qubit |Wi = (|ggei+|gegi+|eggi)/ 3 with fidelity
FW = hW | ρW |Wi = 0.69 ± 0.01. The measured imaginary parts (not shown) are found to be small, with (e) |Im ρψs | < 0.05
and (f) |Im ρW | < 0.06, as expected theoretically.
FW = hW | ρW |Wi = 0.69 ± 0.01, which satisfies the entanglement witness inequality FW > 2/3 for three-qubit
entanglement [28]. Generating either of these classes of
entangled states (bi- and tri-partite) requires only a single entangling operation that is short relative to the characteristic time for two-qubit gates (tg ∼ 50 ns). This
entanglement protocol has the further advantage that it
can be scaled to an arbitrary number of qubits.
The quantum circuit for the compiled version of Shor’s
algorithm is shown in Fig. 3a for factoring the number
N = 15 with a = 4 co-prime [22, 23], which returns the
period r = 2 (“10” in binary) with a theoretical success rate of 50 %. Although the success of the algorithm
hinges on quantum entanglement, the final output is ideally a completely mixed state, σm = (1/2)(|0ih0|+|1ih1|).
Therefore, measuring only the raw probabilities of the
output register does not reveal the underlying quantum
entanglement necessary for the success of the computation. Thus, we perform a runtime analysis with QST
at the three points identified in Fig.3b, in addition to
recording the raw probabilities of the output register.
The first breakpoint in the algorithm verifies the existence of bipartite entanglement. A Bell-singlet |ψs i
is formed after a Hadamard-gate (H) [19] on Q2 and
a Controlled-NOT (CNOT)[7, 13] between Q2 and Q3 .
Figure 3d, is the real part of the density matrix reconstructed from QST on |ψs i. The singlet is formed with
fidelity FBell = hψs | ρBell |ψs i = 0.75 ± 0.01 (|Im ρψs | <
0.05 not shown) and entanglement of formation EOF =
0.43[29].
The algorithm is paused after the second CNOT gate
between Q2 and Q4 to check for tripartite entanglement.
√
At this point a three-qubit |GHZi = (|gggi + |eeei)/ 2,
with fidelity FGHZ = hGHZ| ρGHZ |GHZi = 0.59 ± 0.01
(|Im ρGHZ | < 0.06 not shown) is formed between Q2 , Q3 ,
and Q4 as shown in Fig.3e. This state is found to satisfy
the entanglement witness inequality, FGHZ > 1/2 [28]
indicating three-qubit entanglement.
The third step in the runtime analysis captures all
three qubits at the end of the algorithm, where the final H-gate on Q2 , rotates the three-qubit |GHZi into
|ψ3 i = H2 |GHZi = (|gggi + |eggi + |geei − |eeei)/2. Figure 3f is the real part of the density matrix with fidelity
F = hψ3 | ρ3 |ψ3 i = 0.54±0.01. From the three-qubit QST
we can trace out the register qubit to compare with the
experiment where we measure only the single qubit register and the raw probabilities of the algorithm output.
Ideally, the algorithm returns the binary output“00” or
4
a
Init
Q1 |0>
H
Q2 |0>
H
Modular
Exponentiation
armod(N)
"0"
Quantum
Fourier
Transform
"00" "10"
g
e
1
2 GHZ
H
1/2
1/2
H
Cπ/2
Q3 |0>
0
0
Q4 |0>
1
0
"0"
b
2
1
H
H
ggg
eee
1/2
0
0
3 ψ3
H
Q3
H
d
1
0
1
1/4
"0"
0
1/2
"1"
i
1
ψs
1/2
0
-1/4
eee
ggg
ee
ee
1
H
Cz
gg
1
ggg
f
Q4 |0>
H
"1"
h
0
eee
"1"
Q3 |0>
c Q
2
1
3
"0"
Q2 |0>
"1"
-1/2
eee
0
0
1
0
1
ggg
gg
FIG. 3: Compiled version of Shor’s algorithm. a, Four-qubit circuit to factor N = 15, with co-prime a = 4. The three steps in
the algorithm are initialization, modular exponentiation, and the quantum Fourier transform, which computes ar mod(N) and
returns the period r = 2. b, “Recompiled” three-qubit version of Shor’s algorithm. The redundant qubit Q1 is removed by
noting that HH = I. Circuits a and b are equivalent for this specific case. The three steps of the runtime analysis are labeled
1,2,3. c, CNOT gates are realized using an equivalent controlled-Z (CZ ) circuit. d, Step 1: Bell singlet between √
Q2 and Q3
with fidelity, FBell = hψs | ρBell |ψs i = 0.75 ± 0.01 and EOF = 0.43. e, Step 2: Three-qubit |GHZi = (|gggi + |eeei)/ 2 between
Q2 , Q3 , and Q4 with fidelity FGHZ = hGHZ| ρGHZ |GHZi = 0.59 ± 0.01. f, Step 3: QST after running the complete algorithm.
The three-qubit |GHZi is rotated into |ψ3 i = H2 |GHZi = (|gggi + |eggi + |geei − |eeei)/2 with fidelity, F = 0.55. g,h The
density matrix of the single-qubit output register Q2 formed by: (g), tracing-out Q3 and Q4 from f, and (h) directly measuring
√
√
Q2 with QST, both with F = ρ σm ρ = 0.92 ± 0.01 and SL = 0.78. From 1.5 × 105 direct measurements the output register
returns the period r = 2, with probability 0.483 ± 0.003, yielding the prime factors 3 and 5. (i), The density matrix of the
single-qubit output register without entangling gates, H2 H2 |gi = I |gi. The algorithm fails and returns r = 0 100 % of the
time. Compared to the single quantum state |ψout i = |gi, the fidelity Fcheck = hψg | ρcheck |ψg i = 0.83 ± 0.01, which is less than
unity due to the energy relaxation.
“10” (including the redundant qubit) with equal probability, where the former represents a failure and the latter
indicates the successful determination of r = 2. We use
three methods to analyze the output of the algorithm:
Three-qubit QST, single-qubit QST, and the raw probabilities of the output register state. Figures 3g, h are the
real part of the density matrices for the single qubit output register from three-qubit QST and one-qubit QST
√
√
with fidelity F = ρ σm ρ = 0.92 ± 0.01 for both density matrices. From the raw probabilities calculated from
150,000 repetitions of the algorithm, we measure the output “10” with probability 0.483 ± 0.003, yielding r = 2,
and after classical processing we compute the prime factors 3 and 5.
The linear entropy SL = 4[1 − Tr(ρ2 )]/3[30] is another
metric for comparing the observed output to the ideal
mixed state, where SL = 1 for a completely mixed state.
We find SL = 0.78 for both the reduced density matrix
from the third step of the runtime analysis (three-qubit
QST), and from direct single-qubit QST of the register
qubit.
As a final check of the requisite entanglement, we run
the full algorithm without any of the entangling operations and use QST to measure the single-qubit output
register. The circuit reduces to two H-gates separated by
the time of the two entangling gates. Ideally Q2 returns
to the ground state and the algorithm fails (returns “0”)
100 % of the time. Figure 3i is the real part of the density
matrix for the register qubit after running this check experiment. The fidelity of measuring the register qubit in
|gi is Fcheck = hg| ρcheck |gi = 0.83 ± 0.01. The algorithm
fails, as expected, without the entangling operations.
5
In conclusion, we have implemented a compiled version of Shor’s algorithm on a QuP that correctly finds
the prime factors of 15. We showed that the QuP can
create Bell states, both classes of three-qubit entanglement, and the requisite entanglement for properly executing Shor’s algorithm. In addition, we produce coherent
interactions between four qubits and the bus resonator
with a protocol that can be scaled to create
√ an N -qubit
|Wi state, during which we observe the N dependence
of the effective coupling strength with the number N of
participating qubits. These demonstrations represent an
important milestone for superconducting qubits, further
proving this architecture for quantum computation and
quantum simulations.
Devices were made at the UCSB Nanofabrication Facility, a part of the NSF-funded National Nanotechnology Infrastructure Network. This work was supported
by IARPA under ARO awards W911NF-08-1-0336 and
W911NF-09-1-0375. R.B. acknowledges support from
the Rubicon program of the Netherlands Organisation
for Scientific Research. M.M. acknowledges support from
an Elings Postdoctoral Fellowship. The authors thank
S. Ashhab and A. Galiautdinov for useful comments on
rapid entanglement.
Correspondence
and
requests
for
materials
should be addressed to J.M.M. (email:
[email protected]).
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[1] Shor, P. Algorithms for Quantum Computation : Discrete Logarithms and Factoring. Proceedings of the 35th
Annual Symposium on the Foundations of Computer Science 124–134 (1994).
[2] Vandersypen, L. M. et al. Experimental realization of
Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–7 (2001).
[3] Lanyon, B. et al. Experimental Demonstration of a Compiled Version of Shors Algorithm with Quantum Entanglement. Physical Review Letters 99 (2007).
[4] Lu, C.-Y., Browne, D., Yang, T. & Pan, J.-W. Demonstration of a Compiled Version of Shors Quantum Factoring Algorithm Using Photonic Qubits. Physical Review
Letters 99 (2007).
[5] Politi, A., Matthews, J. C. F. & O’Brien, J. L. Shor’s
quantum factoring algorithm on a photonic chip. Science
325, 1221 (2009).
[6] Clarke, J. & Wilhelm, F. K. Superconducting quantum
bits. Nature 453, 1031–42 (2008).
[7] Mariantoni, M. et al. Implementing the Quantum von
Neumann Architecture with Superconducting Circuits.
Science 334, 61–65 (2011).
[8] Hofheinz, M. et al. Generation of Fock states in a superconducting quantum circuit. Nature 454, 310–4 (2008).
[9] Ansmann, M. et al. Violation of Bell’s inequality in
Josephson phase qubits. Nature 461, 504–6 (2009).
[10] Neeley, M. et al. Generation of three-qubit entangled
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
states using superconducting phase qubits. Nature 467,
570–3 (2010).
Dicarlo, L. et al. Preparation and measurement of threequbit entanglement in a superconducting circuit. Nature
467, 574–8 (2010).
Altomare, F. et al. Tripartite interactions between two
phase qubits and a resonant cavity. Nature Physics 6,
777–781 (2010).
Yamamoto, T. et al. Quantum process tomography of
two-qubit controlled-Z and controlled-NOT gates using
superconducting phase qubits. Physical Review B 82,
1–8 (2010).
Fink, J. et al. Dressed Collective Qubit States and the
Tavis-Cummings Model in Circuit QED. Physical Review
Letters 103, 1–4 (2009).
Steffen, M. et al. State Tomography of Capacitively
Shunted Phase Qubits with High Fidelity. Physical Review Letters 97, 4–7 (2006).
Nielsen, M. A. & Chuang, I. L. Quantum Computation
and Quantum Information (Cambridge University Press,
2000).
Barenco, A. et al. Elementary gates for quantum computation. Physical Review A 52, 3457–3467 (1995).
Lucero, E. et al. High-Fidelity Gates in a Single Josephson Qubit. Physical Review Letters 100, 1–4 (2008).
Lucero, E. et al. Reduced phase error through optimized
control of a superconducting qubit. Physical Review A
82, 1–7 (2010).
DiCarlo, L. et al. Demonstration of two-qubit algorithms
with a superconducting quantum processor. Nature 460,
240–4 (2009).
Tessier, T., Deutsch, I., Delgado, a. & Fuentes-Guridi, I.
Entanglement sharing in the two-atom Tavis-Cummings
model. Physical Review A 68, 1–10 (2003).
Beckman, D., Chari, A., Devabhaktuni, S. & Preskill,
J. Efficient networks for quantum factoring. Physical
Review A 54, 1034–1063 (1996).
Buscemi, F. Shors quantum algorithm using electrons
in semiconductor nanostructures. Physical Review A 83
(2011).
Hofheinz, M. et al. Synthesizing arbitrary quantum
states in a superconducting resonator. Nature 459, 546–9
(2009).
Lucero, E. Computing prime factors on a Josephson
phase-qubit architecture: 15 = 3 ∗ 5 . Ph.D. thesis, University of California Santa Barbara (2012).
Jaynes, E. & Cummings, F. Comparison of quantum
and semiclassical radiation theories with application to
the beam maser. Proceedings of the IEEE 51, 89–109
(1963).
Hill, S. & Wootters, W. Entanglement of a Pair of Quantum Bits. Physical Review Letters 78, 5022–5025 (1997).
Acı́n, a., Bruß, D., Lewenstein, M. & Sanpera, a. Classification of Mixed Three-Qubit States. Physical Review
Letters 87, 2–5 (2001).
We find energy relaxation and dephasing from the qubits
as the primary source of reduced fidelities. T1 ∼ 400ns
and T2 ∼ 200ns for all four qubits and T1 ∼ 3µs for the
bus resonator.
White, A. G. et al. Measuring two-qubit gates. Journal
of the Optical Society of America B 24, 172 (2007).